Front–Concatenation Shuffles on N: Tail Rigidity, Finite–Core Dynamics, and Complete Classifications in Low Rank

This paper analyzes a generalized two-parameter permutation process on the positive integers, termed the Front-Concatenation (FC) (m,p)-Shuffle, inspired by the classic Kimberling problem. We first analyze the fundamental dynamics of the system, proving the Drift Dominance Theorem, which demonstrates that a strong backward drift dominates the dynamics when the re-injection parameter $p>0$, suggeWe study a deterministic two--parameter front--concatenation shuffle on the positive integers.At step n one expels the element in position n of the current infinite list, moves the next m elements to the absolute front, and re--injects the preceding p elements (or fewer when n≤ p) immediately after that forward pull.We prove a tail rigidity theorem: for every $(m,p)$ and every time n, the list agrees exactly with the shifted identity beyond position $n+m-1$, hence the infinite dynamics reduce to a finite recursion on a core of length $n+m-2$.As a main consequence we completely classify the entire $m=1$ family for all p≥ 0: the expelled sequence is Eₙ=2n-1, so the process is never exhaustive and expels precisely the odd integers in increasing order.For $m=2$ we derive a closed core recursion and a distance--to--expulsion dynamics.This yields two universal arithmetic progressions in the expelled sequence for every p.We further give complete, fully proved classifications of the solvable cases $(m,p)=(2,0)$ and $(m,p)=(2,1)$; in particular, $(2,1)$ is exhaustive and its termination mechanism reduces to an explicit ``$3x-1$'' parity cascade.We isolate the remaining obstruction for general $(2,p)$ as a one--dimensional floor--dilation map gᵣ(W)=W+ W/r with $r=p+1$, and prove gᵣ is injective with an explicit inverse; any hypothetical non--exhaustive case would therefore require an explicit infinite gᵣ--orbit avoiding multiples of r. Finally, Lemma~{lem:digit-sensitivity} yields an exact survivor tree for the wrap map: avoiding rZ up to depth T depends only on W rT+1, and there are exactly (r-1)T+1 surviving residue classes. Consequently, under uniform base-r digits the hitting time of rZ is geometric, any non-terminating integers (if they exist) have density $0$, and the r-adic avoidance set is a Cantor-type fractal of Hausdorff dimension log(r-1)/log r.sting exhaustivity across this regime. We then investigate the $m=1, p>0$ family and uncover a stark, parity-driven dichotomy in the structure of the chaos. We provide a rigorous proof for the underlying parity-based mechanism, showing that dynamics exhibit "rapid mixing" for even p and "structured chaos" with long time scales for odd p. Finally, we prove the Collision Lemma, a key combinatorial statement, which resolves the exhaustivity conjecture for the (1,1) FC-shuffle.